Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
553 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 553 conjugacy classes of subgroups — too many to draw. Only its 197 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10,13,16,12,15,20,23)(11,17,19,24,18,14,21,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,21)(23,24);(1,3,4,7)(2,6,5,8)(10,15)(13,20)(14,17)(19,21);(2,5)(6,8)(11,18)(14,17)(19,21)(22,24)Polycyclic presentation:
2^8|a^2=g,c^2=eh,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #139292, #139294, #139320, … |
| maximal normal subgroup of | #139292, #139294, #139320, … |
| maximal quotient of | #222882, #222889, #222899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15469);